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Explicit plot of BesselY0 (red), BesselJ1 (green) and BesselJ0 (blue)
Complex map \(u+\mathrm i v=Y_0(x+\mathrm i y)\)

BesselY0, called also Neumann function, and also \(\mathrm{BesselY}_0\) or simply \(Y_0\) is kind of Bessel function (or Cylindric function), solution \(f=f(z)\) of the Bessel equation

\( \!\!\!\!\!\!\!\!\!\! (1) ~ ~ ~ \displaystyle f''+f'/z+f=0\)

with integral representation

\(\!\!\!\!\!\!\!\!\!\!\!\!\!\! \displaystyle Y_0(x)=\frac{-2}{\pi} \int_0^\infty \cos(x \cosh(t)) \mathrm d t\)

valid at least for \(x>0\).

The explicit plot of \(Y_0\) of real argument is shown at the upper right corner, for comparison, \(J_0\) and \(J_1\) are also plotted; the complex map of \(Y_0\) is shown below.

Various notations

In the literture, and in particular, in the first editions of the tables by Granshtein,Ryzhik, BesselY0 is called Neumann function and denoted \(N_0\). In the latest editions and in the tables by Abramowits,Stegun, notation \(Y_0\) is used. In Mathematica, BesselY0(z) can be accessed with BesselY[0,z] and is interpreted as "Bessel function of Second kind" [1]

Relation to \(J_0\)

In the upper half–plane, id est, for \(\Re(z)>0\), the following relation with BesselJ0 takes place:

\(Y_0(z)=Y_0(-z)+2~ \mathrm i~ J_0(-z)\)

At \(\Re(z)<0\), the similar relation is valid:

\(Y_0(z)=Y_0(-z)- 2~ \mathrm i~ J_0(-z)\)

These formulas are used for evaluation of \(Y_0(z)\) at \(\Re(z)<0\) in the C++ complex(double) implementation Bessely0.cin .

Behavior at zero

\( \!\!\!\!\!\!\!\!\!\! (2) ~ ~ ~ \displaystyle Y_0(z)=\frac{2 (\log (z)+\gamma -\log (2))}{\pi }+\frac{z^2 (-\log (z)-\gamma +1+\log (2))}{2 \pi }+\frac{z^4 (2 \log (z)+2 \gamma -3-2 \log (2))}{64 \pi }+O\left(z^6\right) \)

where \(~\gamma=\)EulerGamma\(~\approx 0.5772156649015329~\) is universal constant.

Expansion at large values of the argument

At large \(|z|\), the expansion of \(Y_0\) can be written in analogy with that for \(J_0\):

\( \!\!\!\!\!\!\!\!\!\!\!\! (3) ~ ~ ~ \displaystyle J_0(z)=\sqrt{\frac{2}{\pi z}} \left( \left(\frac{1}{8 z} -\frac{75}{1024 z^3}+O\left(\frac{1}{z^5}\right)\right) \sin \left(z-\frac{\pi }{4}\right) +\left(1-\frac{9}{128 z^2}+\frac{3675}{32768 z^4}+O\left(\frac{1}{z^6}\right)\right) \cos \left(z-\frac{\pi }{4}\right) \right) \)
\( \!\!\!\!\!\!\!\!\!\!\!\! (4) ~ ~ ~ \displaystyle Y_0(z)=\sqrt{\frac{2}{\pi z}} \left( \left(\frac{-1}{8 z}+\frac{75}{1024 z^3}+O\left(\frac{1}{z^5}\right)\right) \cos \left(z-\frac{\pi }{4}\right) +\left(1-\frac{9}{128 z^2}+\frac{3675}{32768 z^4}+O\left(\frac{1}{z^6}\right)\right) \sin \left(z-\frac{\pi }{4}\right) \right) \)


Bessel function BesselJ0 BesselJ1 BesselH0